SATHEE: Trigonometrical Equations 3 Question 5

Trigonometrical Equations 3 Question 5

5. Number of solutions of the equation $\tan x + \sec x = 2 \cos x$ lying in the interval $[0, 2 π]$ is

(1993, 1M)

(a) 0

(b) 1

(d) 3

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Answer:

Correct Answer: 5. (c)

Solution:

$\tan x + \sec x = 2 \cos x, x \notin (2 n + 1) \frac{π}{2}$

$\begin{aligned} \Rightarrow & \sin x + 1 & = 2 \cos^{2} x \\ \Rightarrow & \sin x + 1 & = 2 (1 - \sin^{2} x) \\ \Rightarrow & 2 \sin^{2} x + \sin x - 1 & = 0 \\ \Rightarrow & (2 \sin x - 1) (\sin x + 1) & = 0 \\ \Rightarrow & \sin x & = \frac{1}{2}, \sin x = - 1 \\ \Rightarrow & x & = \frac{π}{6}, \frac{5 π}{6} \\ x & = \frac{3 π}{2} \\ or & x & \notin (2 n + 1) \frac{π}{2} \\ but & x & = \frac{π}{6}, \frac{5 π}{6} \end{aligned}$

Hence, number of solutions are two.

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Trigonometrical Equations 3 Question 5

5. Number of solutions of the equation tan⁡x+sec⁡x=2cos⁡x lying in the interval [0,2π] is

Answer:

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5. Number of solutions of the equation $\tan x + \sec x = 2 \cos x$ lying in the interval $[0, 2 π]$ is